Subgroups — #26032499

Order 11392.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152257	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151955	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151928	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151922	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151902	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151849	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151842	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151813	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151741	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151736	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13083562	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151676	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151606	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151597	C₂	1	1	yes	maximal, characteristic, complemented
15	#553×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
18	C₂×#13083562	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	C₂×#13083559	C₂²	1	1	yes	characteristic
21	C₂×#13083558	C₂²	1	1	yes	characteristic
22	C₂×#13083558	C₂²	1	1	yes	characteristic
23	#15668000	C₂²	1	1	yes	characteristic
24	#15667998	—	2	1	no	—
25	#15667991	C₂²	1	1	yes	characteristic
26	#15667990	—	2	1	no	—
27	#15667986	C₂²	1	1	yes	characteristic
28	#15667984	—	2	1	no	—
29	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
30	#15667972	—	2	1	no	—
31	#15667945	C₂²	1	1	yes	characteristic
32	#15667943	C₂²	1	1	yes	characteristic
33	#15667943	C₂²	1	1	yes	characteristic
34	C₄⋊#10623494	C₂²	1	1	yes	characteristic
35	#15667935	C₂²	1	1	yes	characteristic
36	#15667935	C₂²	1	1	yes	characteristic
37	#15667935	C₂²	1	1	yes	characteristic
38	#15667935	C₂²	1	1	yes	characteristic
39	#15667935	C₂²	1	1	yes	characteristic
40	#15667935	C₂²	1	1	yes	characteristic
41	#15667933	C₂²	1	1	yes	characteristic
42	#15667932	C₂²	1	1	yes	characteristic
43	#15667932	C₂²	1	1	yes	characteristic
44	#15667932	C₂²	1	1	yes	characteristic
45	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
46	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
47	C₄×#10623494	C₂²	1	1	yes	characteristic
48	#126×C₈₉	—	2	1	no	—
49	#125×C₈₉	C₂²	1	1	yes	characteristic
50	#123×C₈₉	—	2	1	no	—
51	#121×C₈₉	—	2	1	no	—
52	#121×C₈₉	C₂²	1	1	yes	characteristic
53	#121×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₉	—	2	1	no	—
55	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
56	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
57	#117×C₈₉	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	2	2	no	—
60	#13083574	—	4	1	no	—
61	C₂×Dic₁₇₈	—	2	2	no	—
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×Dic₁₇₈	—	4	1	no	—
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
66	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂²×Dic₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	#13083562	—	2	2	no	—
72	#13083562	—	4	1	no	—
73	#13083559	—	4	1	no	—
74	#13083558	—	4	1	no	—
75	#13083558	—	4	1	no	—
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₂×#10623494	C₂³	1	1	yes	characteristic
78	C₂×#10623494	C₂³	1	1	yes	characteristic
79	C₂×#10623494	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	C₂²⋊Dic₈₉	—	2	1	no	—
84	#13083551	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	C₄×Dic₈₉	—	2	1	no	—
88	C₂×D₄×C₈₉	—	2	1	no	—
89	C₂×D₄×C₈₉	—	2	1	no	—
90	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
91	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	—	2	1	no	—
93	C₂²×C₄×C₈₉	—	2	1	no	—
94	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	—	2	1	no	—
97	C₄⋊C₄×C₈₉	—	2	1	no	—
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₈₉	—	2	1	no	—
104	C₂²⋊C₄×C₈₉	—	2	1	no	—
105	C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
108	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
109	C₄²×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂³×C₈₉	—	2	1	no	—
112	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₈₉	—	2	1	no	—
115	D₄×C₈₉	—	2	1	no	—
116	D₄×C₈₉	—	2	1	no	—
117	D₄×C₈₉	—	4	1	no	—
118	D₄×C₈₉	—	4	2	no	—
119	D₄×C₈₉	—	8	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	8	1	no	—
128	C₂×C₄×C₈₉	—	8	1	no	—
129	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
134	C₂²⋊D₈₉	—	4	2	no	—
135	C₂²⋊D₈₉	—	4	2	no	—
136	C₂×Dic₈₉	—	2	1	no	—
137	C₂×Dic₈₉	—	2	1	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	—	4	2	no	—
140	D₃₅₆	—	2	1	no	—
141	D₃₅₆	—	2	1	no	—
142	C₄×D₈₉	—	4	2	no	—
143	Dic₁₇₈	—	2	1	no	—
144	Dic₁₇₈	—	2	1	no	—
145	Dic₁₇₈	—	2	1	no	—
146	Dic₁₇₈	—	4	2	no	—
147	Dic₁₇₈	#41	1	2	yes	—
148	#10623494	—	2	1	no	—
149	#10623494	—	2	1	no	—
150	#10623494	—	2	1	no	—
151	#10623494	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	—	2	1	no	—
154	C₂²×C₈₉	—	2	1	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	—	4	1	no	—
161	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×D₈₉	—	4	1	no	—
163	C₂×D₈₉	—	4	1	no	—
164	C₂×D₈₉	—	4	1	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	—	4	1	no	—
171	C₄×C₈₉	—	4	1	no	—
172	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
173	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
174	Dic₈₉	—	2	2	no	—
175	Dic₈₉	—	4	1	no	—
176	Dic₈₉	—	4	1	no	—
177	C₂×C₈₉	—	4	1	no	—
178	C₂×C₈₉	—	4	1	no	—
179	C₂×C₈₉	—	8	1	no	—
180	C₂×C₈₉	#546	1	1	yes	characteristic
181	C₂×C₈₉	#575	1	1	yes	characteristic
182	C₂×C₈₉	#577	1	1	yes	characteristic
183	D₈₉	—	8	1	no	—
184	#3326	—	89	1	no	maximal, Sylow
185	C₈₉	#3326	1	1	yes	characteristic, complemented, Sylow
186	#553	D₈₉	1	1	yes	characteristic, complemented
187	#548	—	89	1	no	—
188	#488	—	89	1	no	—
189	#483	—	89	1	no	—
190	#482	—	89	1	no	—
191	#479	—	89	1	no	—
192	#469	—	89	1	no	—
193	#468	—	89	1	no	—
194	#463	—	89	1	no	—
195	#449	—	89	1	no	—
196	#448	—	89	1	no	—
197	C₄×SD₁₆	—	89	1	no	—
198	#436	—	89	1	no	—
199	#419	—	89	1	no	—
200	#417	—	89	1	no	—
201	C₂×#41	—	89	1	no	—
202	C₂²×Q₈	—	89	1	no	—
203	C₂×SD₁₆	—	89	1	no	—
204	C₂×SD₁₆	—	89	1	no	—
205	C₂×D₈	—	89	1	no	—
206	C₂×M₁₆	—	89	1	no	—
207	C₂×M₁₆	—	89	1	no	—
208	#126	—	2	1	no	—
209	#125	C₂×D₈₉	1	1	yes	characteristic
210	#124	—	89	1	no	—
211	#124	—	178	1	no	—
212	#123	—	2	1	no	—
213	C₂²⋊Q₈	—	89	1	no	—
214	C₂²⋊Q₈	—	178	1	no	—
215	#121	—	2	1	no	—
216	#121	—	89	1	no	—
217	#121	—	178	1	no	—
218	#121	C₂×D₈₉	1	1	yes	characteristic
219	#121	C₂×D₈₉	1	1	yes	characteristic
220	C₄×Q₈	—	89	1	no	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	—	178	1	no	—
223	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
224	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
225	#117	C₂×D₈₉	1	1	yes	characteristic
226	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
227	#107	—	89	1	no	—
228	#106	—	89	1	no	—
229	#106	—	89	1	no	—
230	C₄⋊C₈	—	89	1	no	—
231	#103	—	89	1	no	—
232	#103	—	89	1	no	—
233	#103	—	89	1	no	—
234	#103	—	89	1	no	—
235	#103	—	89	1	no	—
236	#103	—	89	1	no	—
237	#102	—	89	1	no	—
238	#102	—	89	1	no	—
239	#102	—	89	1	no	—
240	#102	—	89	1	no	—
241	C₂²⋊C₈	—	89	1	no	—
242	C₂²⋊C₈	—	89	1	no	—
243	C₄×C₈	—	89	1	no	—
244	#41	—	356	1	no	—
245	#41	—	178	2	no	—
246	C₂×Q₈	—	89	1	no	—
247	C₂×Q₈	—	89	1	no	—
248	C₂×Q₈	—	356	1	no	—
249	C₂×Q₈	—	178	2	no	—
250	C₂×Q₈	—	178	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	89	1	no	—
254	C₂×D₄	—	178	1	no	—
255	C₂×D₄	—	178	1	no	—
256	C₂×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
257	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	178	1	no	—
261	C₂²×C₄	—	178	1	no	—
262	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
264	SD₁₆	—	356	1	no	—
265	SD₁₆	—	178	2	no	—
266	D₈	—	356	1	no	—
267	M₁₆	—	356	1	no	—
268	M₁₆	—	356	1	no	—
269	C₂×C₈	—	89	1	no	—
270	C₂×C₈	—	89	1	no	—
271	C₂×C₈	—	89	1	no	—
272	C₂×C₈	—	89	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	89	1	no	—
277	C₄⋊C₄	—	178	1	no	—
278	C₄⋊C₄	—	178	1	no	—
279	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
280	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
281	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	2	1	no	—
287	C₂²⋊C₄	—	178	1	no	—
288	C₂²⋊C₄	—	178	1	no	—
289	C₂²⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	—	178	1	no	—
291	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
292	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
293	C₄²	—	2	1	no	—
294	C₄²	—	178	1	no	—
295	C₄²	C₂²×D₈₉	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	178	1	no	—
298	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
299	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
300	Q₈	—	89	2	no	—
301	Q₈	—	356	2	no	—
302	Q₈	—	178	1	no	—
303	Q₈	—	178	1	no	—
304	Q₈	—	178	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	2	no	—
309	D₄	—	8	1	no	—
310	D₄	—	356	2	no	—
311	D₄	—	356	2	no	—
312	D₄	—	178	1	no	—
313	D₄	—	178	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	8	1	no	—
322	C₂×C₄	—	8	1	no	—
323	C₂×C₄	—	356	2	no	—
324	C₂×C₄	—	356	2	no	—
325	C₂×C₄	—	178	1	no	—
326	C₂×C₄	—	178	1	no	—
327	C₂×C₄	—	178	1	no	—
328	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
330	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
331	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
333	C₈	—	178	1	no	—
334	C₈	—	178	1	no	—
335	C₈	—	178	1	no	—
336	C₈	—	178	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	356	1	no	—
347	C₂²	—	356	1	no	—
348	C₂²	—	356	1	no	—
349	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	356	1	no	—
358	C₄	—	356	1	no	—
359	C₄	—	178	2	no	—
360	C₄	#41×D₈₉	1	1	yes	characteristic
361	C₄	#15668058	1	1	yes	characteristic
362	C₂	—	4	1	no	—
363	C₂	—	4	1	no	—
364	C₂	—	8	1	no	—
365	C₂	—	712	1	no	—
366	C₂	#19152236	1	1	yes	characteristic
367	C₂	#19152396	1	1	yes	characteristic
368	C₂	#19152409	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#553×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#553	D₈₉
hypocenter	C₈₉	#3326
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #553

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #19152257	C₂	semidirect
2. #19151955	C₂	semidirect
3. #19151928	C₂	semidirect
4. #19151922	C₂	semidirect
5. #19151902	C₂	semidirect
6. #19151849	C₂	semidirect
7. #19151842	C₂	semidirect
8. #19151813	C₂	semidirect
9. #19151741	C₂	semidirect
10. #19151736	C₂	semidirect
11. C₄×#13083562	C₂	semidirect
12. #19151676	C₂	semidirect
13. #19151606	C₂	semidirect
14. #19151597	C₂	semidirect
15. #553×C₈₉	C₂	semidirect
185. C₈₉	#3326	semidirect
186. #553	D₈₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19152257	2
#19151955	2
#19151928	2
#19151922	2
#19151902	2
#19151849	2
#19151842	2
#19151813	2
#19151741	2
#19151736	2
C₄×#13083562	2
#19151676	2
#19151606	2
#19151597	2
#553×C₈₉	2
#3326	89